health problem vs ill health

health problem and ill health both are nouns.

Word	Noun	Adjective	Verb	Adverb
health problem	Yes	No	No	No
ill health	Yes	No	No	No

As nouns, health problem and ill health are synonyms defined as:

health problem and ill health: a state in which you are unable to function normally and without pain

Other synonyms of health problem include unhealthiness.

health problem (noun)	ill health (noun)
a state in which you are unable to function normally and without pain	a state in which you are unable to function normally and without pain